arXiv subjects

Maxime Ingremeau

Publications and source records attributed to Maxime Ingremeau.

At least 19 recordsLinked to original sources

arXiv2024

Improved $L^\infty$ bounds for eigenfunctions under random perturbations in negative curvature

It has been known since the work of Avakumov\'ic, H\"ormander and Levitan that, on any compact smooth Riemannian manifold, if $-\Delta_g \psi_\lambda = \lambda \psi_\lambda$, then $\|\psi_\lambda\|_{L^\infty} \leq C \lambda^{\frac{d-1}{4}} \|\psi_\lambda\|_{L^2}$. It is believed that, on manifolds of negative curvature, such a bound can be largely improved; however, only logarithmic improvements in $\lambda$ have been obtained so far. In the present paper, we obtain polynomial improvements over the previous bound in a generic setting, by adding a small random pseudodifferential perturbation to the Laplace-Beltrami operator.

math.SP↗

arXiv2023

Emergence of Gaussian fields in noisy quantum chaotic dynamics

We study the long time Schr\"odinger evolution of Lagrangian states $f_h$ on a compact Riemannian manifold $(X,g)$ of negative sectional curvature. We consider two models of semiclassical random Schr\"odinger operators $P_h^\alpha=-h^2\Delta_g +h^\alpha Q_\omega$, $0<\alpha\leq 1$, where the semiclassical Laplace-Beltrami operator $-h^2\Delta_g$ on $X$ is subject to a small random perturbation $h^\alpha Q_\omega$ given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of $Q_\omega$ is bounded, but oscillates and decorrelates at scale $h^{\beta}$, $0< \beta < \frac{1}{2}$. We prove a quantitative result that, under appropriate conditions on $\alpha,\beta$, in probability with respect to $\omega$ the long time propagation $$\mathrm{e}^{\frac{i}{h}t_h P_h^\alpha } f_h, \quad o(|\log h|)=t_h\to\infty, ~~h\to 0,$$ rescaled to the local scale of $h$ around a uniformly at random chosen point $x_0$ on $X$, converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and $\omega$-almost sure version of this convergence along sufficiently fast decaying subsequences $h_j\to 0$.

math.AP↗

arXiv2022

A trace formula for scattering resonances of unbalanced quantum graphs

Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals outside a strip. We deduce lower bounds on the number of resonances (in bounded regions of the complex plane),that are independent of the size of the graph. We also deduce partial results indicating that Benjamini-Schramm convergence of open quantum graphs should imply convergence of the empirical spectral measures.

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arXiv2021

How Lagrangian states evolve into random waves

In this paper, we consider a compact manifold $(X,d)$ of negative curvature, and a family of semiclassical Lagrangian states $f_h(x) = a(x) e^{\frac{i}{h} ϕ(x)}$ on $X$. For a wide family of phases $ϕ$, we show that $f_h$, when evolved by the semiclassical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry's random waves conjecture for Lagrangian states.

math-ph↗

arXiv2021

Scattering resonances of large weakly open quantum graphs

In this paper, we consider a sequence of open quantum graphs, with uniformly bounded data, and we are interested in the asymptotic distribution of their scattering resonances. Supposing that the number of leads in our quantum graphs is small compared to the total number of edges, we show that most resonances are close to the real axis. More precisely, the asymptotic distribution of resonances of our open quantum graphs is the same as the asymptotic distribution of the square-root of the eigenvalues of the closed quantum graphs obtained by removing all the leads.

math.SP↗

arXiv2021

Semiclassical limits of distorted plane waves in chaotic scattering without a pressure condition

In this paper, we study the semi-classical behavior of distorted plane waves, on manifolds that are Euclidean near infinity or hyperbolic near infinity, and of non-positive curvature. Assuming that there is a strip without resonances below the real axis, we show that distorted plane waves are bounded in $L^2_{loc}$ independently of $h$, that they admit a unique semiclassical measure, and we prove bounds on their $L^p_{loc}$ norms.

math.SP↗

arXiv2021

Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization

We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in the sense that their spectrum in $I$ is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in $I$ are spatially delocalized.

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arXiv2020

Local Weak Limits of Laplace Eigenfunctions

In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent results of Bourgain, Buckley and Wigman, we will prove that some deterministic families of eigenfunctions on $\mathbb{T}^2$ satisfy the conclusions of the random wave conjecture. We also show that on an arbitrary domain, a sequence of Laplace eigenfunctions always admits local weak limits. We explain why these local weak limits can be a powerful tool to study the asymptotic number of nodal domains.

math.AP↗

arXiv2020

Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit

We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a randomly chosen ball has a limiting distribution. We prove that any sequence of quantum graphs with uniformly bounded data has a convergent subsequence in this sense. We then consider the empirical spectral measure of a convergent sequence (with general boundary conditions and edge potentials) and show that it converges to the expected spectral measure of the limiting random rooted quantum graph. These results are similar to the discrete case, but the proofs are significantly different.

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arXiv2020

Absolutely Continuous Spectrum for Quantum Trees

We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates.

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arXiv2019

Quantum ergodicity for large equilateral quantum graphs

Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant $α$) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case $α$ = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit analytic density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

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arXiv2019

Equidistribution of Phase Shifts in Obstacle Scattering

For scattering off a smooth, strictly convex obstacle $Ω\subset \mathbb{R}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix -- the phase shifts -- equidistribute on the unit circle as the frequency $k \to \infty$ at a rate proportional to $k^{d - 1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S \subset \mathbb{S}^1$ not containing $1$, there are $c_d |S| \mathrm{Vol}(\partial Ω)\ k^{d - 1} + o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ is a constant depending only on the dimension. Using this result, the two term asymptotic expansion for the counting function of Dirichlet eigenvalues, and a spectral-duality result of Eckmann-Pillet, we then give an alternative proof of the two term asymptotic of the total scattering phase due to Majda-Ralston.

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arXiv2018

The semi-classical scattering matrix from the point of view of Gaussian states

In this note, we will consider semiclassical scattering for compactly supported non-trapping perturbations of the Laplacian on $\mathbb{R}^d$. We will define a family of Gaussian states on $\mathbb{S}^{d-1}$, parametrized by points in $T^*\mathbb{S}^{d-1}$, and show that the action of the scattering matrix on a Gaussian state of parameter $ρ\in T^*\mathbb{S}^{d-1}$ is still a Gaussian state, with parameter $κ(ρ)$, where $κ$ is the (classical) scattering map. This is one way of saying that \emph{the scattering matrix quantizes the scattering map}, complementary to a previous result of Alexandrova in terms of Fourier Integral Operators.

math.AP↗

arXiv2018

A lower bound for the Bogomolny-Schmit constant for random monochromatic plane waves

This note deals with nodal domains of random monochromatic plane waves. It was shown by Nazarov and Sodin that the expected number of such nodal domains included in a disk of radius $R$ is proportional to $πR^2$ in the large $R$ limit. However, very little is known on the value of the proportionality constant from a mathematical point of view. The aim of this note is to obtain a lower bound on the value of this constant my elementary means.

math-ph↗

arXiv2017

Sharp resolvent bounds and resonance-free regions

In this note, we consider semiclassical scattering on a manifold which is Euclidean near infinity or asymptotically hyperbolic. We show that, if the cut-off resolvent satisfies polynomial estimates in a strip of size $O(h |\log h|^{-α})$ below the real axis, for some $α\geq 0$, then the cut-off resolvent is actually bounded by $O(|\log h|^{α+1} h^{-1})$ in this strip. As an application, we improve slightly the estimates on the real axis given by Bourgain and Dyatlov in the case of convex co-compact surfaces.

math.SP↗

arXiv2016

Lower bounds for the number of nodal domains for sums of two distorted plane waves in non-positive curvature

In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees an optimal lower bound, of the same order as Courant's upper bound. As an application, we will obtain optimal lower bounds for the number of nodal domains of distorted plane waves on some families of surfaces of non-positive curvature.

math-ph↗

arXiv2016

Distorted plane waves on manifolds of nonpositive curvature

We will consider the high frequency behaviour of distorted plane waves on manifolds of nonpositive curvature which are Euclidean or hyperbolic near infinity, under the assumption that the curvature is negative close to the trapped set of the geodesic flow and that the topological pressure associated to half the unstable Jacobian is negative. We obtain a precise expression for distorted plane waves in the high frequency limit, similar to the one in \cite{GN} in the case of convex co-compact manifolds. In particular, we will show $L_{loc}^\infty$ bounds on distorted plane waves that are uniform with frequency. We will also show that the real part of distorted plane waves restricted to a compact set satisfy the analogue of Yau's conjecture about the Haussdorff measure of nodal sets.

math-ph↗

arXiv2016

Equidistribution of phase shifts in trapped scattering

We prove an equidistribution result for the eigenvalues of the scattering matrix associated to an operator of the form $-h^2Δ+ V-1$, where $V\in C_c^\infty(\mathbb{R}^d)$ is a compactly supported potential, under the assumption that the incoming and outgoing sets of the classical dynamics have zero Liouville measure. This extends a recent result of Gell-Redman, Hassell and Zelditch, where the authors proved equidistribution of the eigenvalues of the scattering matrix under the assumption that the trapped set is empty.

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